## Abstract

This article introduces a new class of location estimators by applying an existing

measure of location to a set of multiscale means computed on the order statistics of a data set.

Class members are shown to have interesting and desirable properties such as: the member using

the geometric mean is always sandwiched between the geometric and arithmetic means.

The member using the median reduces to a simple form, the guard estimator,

which is the median of three quantities: the sample mean and two guard values based on two or four order

statistics nearest the median. The guard estimator is unbiased, consistent (at least for symmetric distributions),

computable in linear time, does not require tuning parameters and simulations suggest that

it achieves high efficiency: almost matching the mean or median's better efficiency under different distributions.

Guard's finite sample breakdown point demonstrates that it is highly robust even for small samples

and matches the median's breakdown value asymptotically. Two examples exhibit the new location

measures in action: one provides confirmation of a robust approach to establishing whether

Shoshoni leather goods were designed to the `Golden Ratio' standard; the other compares

four functional measures of location for the Aberporth wind speed series.

The new class of location estimators is inspired by the member that uses the geometric mean which

arises naturally from a theoretical analysis of multiscale variance stabilization (MVS) techniques.

The article

introduces maximum likelihood approaches for MVS techniques for independently and

identically distributed data

and sheds theoretical light on MVS by presenting analytical formulae for their Jacobians, a key component

of the likelihood.

The MVS techniques are shown empirically to compare favourably to the well-known Box-Cox

transform, but do not dominate it.

measure of location to a set of multiscale means computed on the order statistics of a data set.

Class members are shown to have interesting and desirable properties such as: the member using

the geometric mean is always sandwiched between the geometric and arithmetic means.

The member using the median reduces to a simple form, the guard estimator,

which is the median of three quantities: the sample mean and two guard values based on two or four order

statistics nearest the median. The guard estimator is unbiased, consistent (at least for symmetric distributions),

computable in linear time, does not require tuning parameters and simulations suggest that

it achieves high efficiency: almost matching the mean or median's better efficiency under different distributions.

Guard's finite sample breakdown point demonstrates that it is highly robust even for small samples

and matches the median's breakdown value asymptotically. Two examples exhibit the new location

measures in action: one provides confirmation of a robust approach to establishing whether

Shoshoni leather goods were designed to the `Golden Ratio' standard; the other compares

four functional measures of location for the Aberporth wind speed series.

The new class of location estimators is inspired by the member that uses the geometric mean which

arises naturally from a theoretical analysis of multiscale variance stabilization (MVS) techniques.

The article

introduces maximum likelihood approaches for MVS techniques for independently and

identically distributed data

and sheds theoretical light on MVS by presenting analytical formulae for their Jacobians, a key component

of the likelihood.

The MVS techniques are shown empirically to compare favourably to the well-known Box-Cox

transform, but do not dominate it.

Original language | English |
---|---|

Pages | 1 |

Number of pages | 39 |

Publication status | Unpublished - 2013 |

## Keywords

- location measure
- guard estimator
- multimeans
- geometric mean
- median
- breakdown
- variance stabilization
- Haar-Fisz
- Haar wavelet